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<h1 class="reftitle">intersect</h1>
<h2>Purpose</h2>
<p>Intersect two polyhedra.</p>
<h2>Syntax</h2>
<pre class="synopsis">PnS = intersect(P, S)</pre>
<h2>Description</h2>
<p></p>
	  Compute the intersection of two polyhedra:
 
	  <p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect5.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect5.png"></p>

	  Note: If <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect1.png"> or <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect2.png"> are in V-rep, then this function will first compute their convex
	  hull.
	<h2>Input Arguments</h2>
<table cellspacing="0" class="body" cellpadding="4" border="0" width="100%">
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<col width="31%">
<col width="69%">
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<td><tt>P</tt></td>
<td>
<p></p>Polyhedron in any format<p>
	    		Class: <tt>Polyhedron</tt></p>
</td>
</tr>
<tr valign="top">
<td><tt>S</tt></td>
<td>
<p></p>Polyhedron in any format<p>
	    		Class: <tt>Polyhedron</tt></p>
</td>
</tr>
</tbody>
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<h2>Output Arguments</h2>
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<td><tt>PnS</tt></td>
<td>
<p></p>Intersection of <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect3.png"> and <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect4.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect4.png">.<p>
	    		Class: <tt>Polyhedron</tt></p>
</td>
</tr></tbody>
</table>
<h2>Example(s)</h2>
<h3>Example 
				1</h3>Intersect two polytopes.<pre class="programlisting">P = ExamplePoly.randVrep;</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">S = ExamplePoly.randHrep;</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">PnS = intersect(P,S)</pre>
<pre class="programlisting">Polyhedron in R^2 with representations:
    H-rep (redundant)   : Inequalities  15 | Equalities   0
    V-rep               : Unknown (call computeVRep() to compute)
Functions : none
</pre>
<pre class="programlisting">plot([P;S],'alpha',0.1);hold on; 
	    plot(PnS,'color','r','alpha',0.6);</pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_1.png" width="60%"></p>
<h3>Example 
				2</h3>Intersect a polytope and an unbounded polyhedron.<pre class="programlisting">P = Polyhedron('V', [1 1;0 1;1 0], 'R', -[1 0]);</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">S = Polyhedron('H', [randn(10,2) 4*ones(10,1)]);</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">PnS = intersect(P,S)</pre>
<pre class="programlisting">Polyhedron in R^2 with representations:
    H-rep (redundant)   : Inequalities  13 | Equalities   0
    V-rep               : Unknown (call computeVRep() to compute)
Functions : none
</pre>
<pre class="programlisting">plot(P,'color','b','alpha',0.1);hold on;
	       plot(S,'color','g','alpha',0.1); plot(PnS,'color','r','alpha',0.6);</pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_2.png" width="60%"></p>
<h3>Example 
				3</h3>Intersect a 3D polytope and a lower dimensional cone.<pre class="programlisting">P = Polyhedron('lb',-0.5*ones(3,1), 'ub', 0.5*ones(3,1));</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">S = Polyhedron('R', [-1 0 .5;-1 2 2]);</pre>
<pre class="programlisting"></pre>
<pre class="programlisting">PnS = intersect(P,S)</pre>
<pre class="programlisting">Polyhedron in R^3 with representations:
    H-rep (redundant)   : Inequalities   8 | Equalities   1
    V-rep               : Unknown (call computeVRep() to compute)
Functions : none
</pre>
<pre class="programlisting">plot(P,'color','b','alpha',0.1);hold on;
	       plot(S,'color','g','alpha',0.1); plot(PnS,'color','r','alpha',1);</pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/intersect_img_3.png" width="60%"></p>
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<br><p>©  <b>2010-2013</b>     Colin Neil Jones: EPF Lausanne,    <a href="mailto:colin.jones@epfl.ch">colin.jones@epfl.ch</a></p>
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